Found problems: 313
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.
Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and
max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.
Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$
Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.
Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$
Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$
Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral.
[i](tatari/nightmare)[/i]
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions:
(i) $f(x)$ is an integer if and only if $x$ is an integer;
(ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that
$f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$.
by Athanasios Kontogeorgis (aka socrates)
The function $g$, with domain and real numbers, fulfills the following:
$\bullet$ $g (x) \le x$, for all real $x$
$\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$
Find $g (1990)$.
Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$
(Folklore)
[hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]''
I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url]
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Let $\mathbb{N}_{\ge 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$:
(a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$,
(b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$,
(c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.
Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.
(a) Prove that for every $x \in R$ holds that
$$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$
(b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that
$$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$
Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.
Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.